8 players in round robin

) Dave Gates' encouragement makes me post a new puzzle. ) ) Eight persons play in round robin. (right expression? sorry about my poor ) English) ) ) In this game player A and B were common winners. ) ) But the rivals C and D made a curious record. ) ) Player C defeated all players who defeated player D and ) player D defeated all players who defeated player C. ) ) Then what is A's score? ) ) There was no draw in this game.

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There are 28 games played in a round-robin tournament, all of which were won by someone, so there are 28 wins to be divided amongst the players.

There has to have been a loser in the game A vs. B, and both A and B were defeated by either C or D. So the maximum number of wins they can have is 5. (they played 7 games and lost at least 2)

You can't have only 2 players win an 8-player round robin with only 4 wins, as there would be 20 wins left, to be divided over 6 players, which averages more than 3 per player.

So players A and B both won 5 out of 7 games.

I came to the same conclusion slightly differently: Either C or D must have scored 4, so A & B must have scored 5 to come out ahead of them.

I think this should be a good puzzle, but I need some clarification please...

Question 1: What is a 'round robin?' Are players eliminated? Or does each player get a chance to play all the other players? In other words, is it safe to say that A played 7 games, B played 7 games, C played 7 games, etc.?

Question 2: What are 'common winners?' Does that mean they tied for first place?

Question 3: Do you mean that whenever two people played a single game, there was never a draw? Or that when we look at the scores, no two people got the same total? (A won 7 games, B won 6 games, C won 5 games, etc.)

Thanks in advance!

Always impatient, I decided to make my own assumptions about what the rules meant, and I was happy with the answer I got, so I think I guessed ok.

etc.?

Interpretation 1: Round Robin means that every player played 7 games. Each player got to face each other player once.

Interpretation 2: This was the tricky one. But I'm guessing it means they did in fact tie for first place.

Interpretation 3: No two players had a draw in a single game. However, there was a tie at the end when A and B both won the same number of games.

Okay, based on this...

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STEP 1. There were no players who beat both C and D. Therefore, between the two of them, C and D beat at least 7 players. Therefore, one of them won at LEAST 4 games.

Therefore... STEP 2. Players A and B had to win more than 4 games.

STEP 3. There had to be at least 3 losses between A and B. At least one of them lost to C and at least one of them lost to D. Plus, one of them lost to the other one.

Therefore... STEP 4. A and B didn't win 7 games each. They didn't even win 6 games each (since that would only mean 2 losses between them).

STEP 5. A and B won less than 6 games and more than 4 games...

So...

STEP 6. Solution: Player A won 5 games.

Park Poo-sung,

Assuming I got this correct, this was another very nice puzzle. Brand new, as far as I know, and the way the clue about C and D doubled back to be used in both STEP 1 and STEP 3 was very nice.

Thanks again!

My answers are below your questions.

Thanks for your praise but sorry about my poor English.

You're right. Each played 7 games. They were not eliminated.

But it is a little trap not to say exaclty '7 games'.

Some people guess that the answer is 6 wins and 2 loses.

Of course, people at this newsgroup are too smart not to be entraped.

I got the hint for this little puzzle from the system of the professional go(baduk in Korea) game.

In most professional go games, 8 players who get through the elimination matches play in round-robin tournament.

Then the player of best score have a right to challenge the title holder.

If two or more players have the same score, they must fight again in order to choose a challenger.

There's no rule in this system that a plyer who defeated a player of same score is considered a winner. (What is the term meaning this rule? In Korea this rule is called 'seung ja seung', 'seung' means 'win' and 'ja' means 'person', so it means 'the winner wins'.)

By the way, what I want is just that two winners A and B have the same score, not who is the winner.

Right, a single game, not scores.

Many puzzles of this type need to have no draw.

Sorry about my poor English again.

In U.S. football, the closest term I can think of is 'head-to-head tiebreaker'.

(I'm covering both NFL and collegiate football here) Usually a group of anywhere from 6 to 15 teams is grouped into a conference; an unbalanced schedule is played, and the team with the most wins is the winner, advancing to some playoff (against winners of other conferences) or prize game (called 'bowl games'.

If two (or more) teams in a conference are tied in number of wins, a series of tiebreakers is invoked to determine the inner of the conference. The first tiebreaker is usually how they did when they played each other, which is called 'head-to-head'.